Calculus Maximus WS 5.2: Slope Fields - korpisworldkorpisworld.com/Mathematics/Calculus...

Calculus Maximus WS 5.2: Slope Fields

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Name_________________________________________ Date________________________ Period______ Worksheet 5.2—Slope Fields Show all work when applicable. Short Answer and Free Response: Draw a slope field for each of the following differential equations.

1. 1dy xdx

�

2. 2dy ydx

3. dy x ydx

�

4. 2dy xdx

5. 1dy ydx

�

6. dy ydx x

�


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For 7 – 12, match each slope field with the equation that the slope field could represent. 7.

8.

9.

10.

11.

12.

(A) 1y (B) y x (C) 2y x

(D) 316

y x (E) 21yx

(F) siny x

(G) cosy x (H) lny x


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For 13 – 16, match the slope fields with their differential equations. 13.

14.

15.

16.

(A) 1 12

dy xdx

�

(B) dy x ydx

�

(C) dy ydx

(D) dy xdx y

�

17. The calculator-drawn slope field for the differential equation dy x ydx

� is shown in the figure below.

(a) Sketch the solution curve through the point � �0,1 .

(b) Sketch the solution curve through the point � �3,0� .

(c) Approximate ( 3.1)y � using the equation of the tangent line to ( )y f x at the point � �3,0� .


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18. Consider the differential equation 2 4dy y xdx

� .

(a) The slope field for the differential equation is shown below. Sketch the solution curve that passes through the point � �0,1 and sketch the solution curve that goes through the point � �0, 1� .

(b) There is a value of b for which 2y x b � is a solution to the differential equation. Find this value

of b. Justify your answer.

(c) Let g be the function that satisfies the given differential equation with the initial condition � �0 0g .

It appears from the slope field that g has a local maximum at the point � �0,0 . Using the differential equation, prove analytically that this is so.


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Multiple Choice: 19. Given the following slope field (with equilibrium

solutions, meaning the slopes are zero and a the existence of a horizontal asymptote on the solution graph, at 0y and 1y ), find the matching differential equation.

(A) � �1dy y ydx

� (B) � �1dy y ydx

�

(C) � �

11

dydx y y

�

(D) � �11 y ydy edx

� �

(E) 1

dy ydx y

�

20. A slope field for the differential equation 42dy ydx

� will show

(A) a line with slope 1� and y-intercept of 42 (B) a vertical asymptote at 42x (C) a horizontal asymptote at 42y (D) a family of parabolas opening downward

(E) a family of parabolas opening to the left 21. For which of the following differential equations will a slope field show nothing but negative slopes in

the fourth quadrant?

(A) dy xdx y

� (B) 5dy xydx

� (C) 2 2dy xydx

� (D) 3

2dy xdx y

(E) 2 3dy ydx x

�

22. Which of the following differential equations would produce the slope field shown below?

(A) dy y x

dx � (B) dy y x

dx � (C) dy y x

dx � (D) dy y x

dx � (E) dy y x

dx �


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23. Which of the following differential equations would produce the slope field shown below?

(A) 3dy y x

dx � (B)

3dy xydx

� (C) 3

dy xydx

� (D) 3

dy yxdx

� (E) 3

dy yxdx

�

24. AP 2010B-5 (No Calculator)

Consider the differential equation 1dy xdx y

� .

(a) On the axes provided at right, sketch a slope field for the given differential equation at the twelve points indicated, and for

1 1x� � � , sketch the solution curve that passes through the point � �0, 1� .

(b) While the slope field in part (a) is drawn at only twelve points, it is

defined at every point in the xy-plane for which 0y z . Describe all

points in the xy-plane, 0y z , for which 1dydx

� .

(c) Find the particular solution � �y f x to the given differential equation with the initial condition

� �0 2f � .


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25. AP 2006-5 (No Calculator)

Consider the differential equation 1dy ydx x

� , where 0x z .

(a) On the axes provided, sketch a slope field for the given differential equation at the eight points

indicated.

(b) Find the particular solution � �y f x to the differential equation with the initial condition

� �1 1f � and state its domain.


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26. AP 2005-6 (No Calculator)

Consider the differential equation 2dy xdx y

� .

(a) On the axes provided, sketch a slope field for the given differential equation at the twelve points indicated.

(b) Let � �y f x be the particular solution to the differential equation with the initial condition

� �1 1f � . Write an equation for the line tangent to the graph of f at � �1, 1� and use it to

approximate � �1.1f .

(c) Find the particular solution � �y f x to the given differential equation with the initial condition

� �1 1f � .

Calculus Maximus WS 5.2: Slope Fields - korpisworldkorpisworld.com/Mathematics/Calculus...

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